Structural Analysis Fundamentals / Complex stress and strain

Strain gauges and yield criteria

Measuring strain on a real structure, and deciding whether the material will yield.

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Why this matters

You cannot measure stress. You can only measure strain, on a surface, with a gauge. Turning three gauge readings into principal stresses is one of the most practical skills in this whole subject, and it is how load tests and monitoring schemes actually work.

By the end of this lesson you should be able to

Explain why three gauges are needed
Convert rosette readings into strain components
Find principal strains and hence principal stresses
Apply the Tresca and von Mises yield criteria

A strain gauge measures the direct strain along its own axis and nothing else. The in-plane strain state has three unknowns — εx, εy and γxy — so you need three independent readings. That is what a rosette is: three gauges at known angles on the same spot.

A 45° rosette places them at 0°, 45° and 90°. A 60° delta rosette places them at 0°, 60° and 120°. Both give three equations for the three unknowns.

45° rosette

\varepsilon_x = \varepsilon_a, \qquad \varepsilon_y = \varepsilon_c, \qquad \gamma_{xy} = 2\varepsilon_b - \varepsilon_a - \varepsilon_c

What it calculates: The in-plane strain components from three gauges at 0°, 45° and 90°.

εa: Gauge at 0°
εb: Gauge at 45°
εc: Gauge at 90°
γxy: Engineering shear strain

This assumes

The gauges are on a free surface, so the stress perpendicular to it is zero
The strain field is uniform over the gauge area

In plain terms: The first two are immediate — a gauge along x reads εx. The third comes from writing the strain transformation at 45° and rearranging. Note that shear strain cannot be measured directly; it is always inferred.

Once the strain components are known, the principal strains follow from a circle exactly like Mohr's circle for stress, but with γ/2 on the vertical axis. Then Hooke's law for plane stress converts the principal strains into principal stresses.

Plane-stress Hooke's law

\sigma_1 = \frac{E}{1-\nu^2}(\varepsilon_1 + \nu\varepsilon_2), \qquad \sigma_2 = \frac{E}{1-\nu^2}(\varepsilon_2 + \nu\varepsilon_1)

What it calculates: Principal stresses from measured principal strains on a free surface.

E: Young's modulus (N/mm²)
ν: Poisson's ratio
ε1, ε2: Principal strains

This assumes

Plane stress (free surface)
Linear elastic, isotropic material

In plain terms: The Poisson terms matter. A gauge stretching in one direction is partly caused by stress in the perpendicular direction, so you cannot simply multiply each strain by E.

Worked example

From rosette readings to principal stresses

Given

A 45° rosette on a steel surface reads εa = 400 μ, εb = 250 μ, εc = −100 μ
E = 205 000 N/mm², ν = 0.30
(1 μ = 1 microstrain = 10⁻⁶)

Find

The principal strains and the principal stresses.

Assumptions

Free surface, so plane stress applies
Linear elastic isotropic steel

The last question is whether the material will yield. A tensile test gives one number — the yield stress — but a real point carries two or three principal stresses at once. A yield criterion reduces the combined state to a single equivalent stress that can be compared with that one test result.

Tresca and von Mises criteria (plane stress)

\sigma_{Tresca} = \max(|\sigma_1-\sigma_2|,|\sigma_1|,|\sigma_2|), \qquad \sigma_{vM} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}

What it calculates: An equivalent uniaxial stress to compare with the yield strength.

σ1, σ2: Principal stresses (N/mm²)
σTresca: Equivalent stress on the maximum-shear criterion (N/mm²)
σvM: Equivalent stress on the shear-strain-energy criterion (N/mm²)

This assumes

Ductile material that yields by shearing
Plane stress: include the zero out-of-plane principal stress when finding the largest principal-stress difference. For general 3D stress, use all three principal stresses.

In plain terms: Tresca says yielding starts when the maximum shear stress reaches its value in a tensile test. Von Mises says it starts when the shear strain energy does. Tresca is the more conservative of the two; von Mises agrees better with tests on metals.

Practice

At a point the principal stresses are σ1 = 150 N/mm² and σ2 = −60 N/mm². What is the Tresca equivalent stress?

Practice

For that same point, what is the von Mises equivalent stress?

Check yourself

Why does a strain rosette need three gauges?

Summary

A gauge measures direct strain along its own axis only, so three are needed
45° rosette: εx = εa, εy = εc, γxy = 2εb − εa − εc
Principal strains come from a circle just like Mohr's, with γ/2 vertical
Plane-stress Hooke's law converts principal strains to principal stresses — the Poisson terms matter
Tresca and von Mises reduce a combined stress state to one equivalent value; Tresca is more conservative